I agree is clear that FR has a lot of support offices in Italy, but the most common interpretation so far is that the nationality of the company is determined by where the main headquarter is based, where the administrator operate and take their decisions. I am pretty sure it is in Ireland in this case, but you never know of course.
Let's take a company with bases worldwide. It would be difficult and very expensive to pay taxes in every single country you have a base, with all the different legislations.
Or, like many, including myself, you can get the gross salary in your bank account, and once or twice a year pay the taxes where you are due to pay. Income taxes is after all on the worker, not on the company. In most circumstances the employer can pay the taxes on your behalf taking it from your payslip, but in all the country as far as I know there are provisions to leave this task to the worker, through a commercialist or by himself.
How to pay social security?
I know how it works in Italy for foreign companies.
If you have many employees the most convenient is to have an office, or even a company employee in Italy, that in from of Italian SS system acts on behalf of the employer, paying contributions and adhering to the local laws.
Smaller companies achieve the same result by delegating an Italian commercialist that will do the same things, paying the employer and the employee part of social security and everything else. It is not as difficult as it sounds in fact, and I am pretty sure that the same is in all the other countries...
In my case it is a Maltese company with employees in all Europe.